The energy dependence of \(\bar{E}_p/E_i\) , as well as the increase of the cross section and pion multiplicity with energy, also implies that the resulting neutrino spectral shape will slightly differ from that of the original proton spectrum. While the first effect would tend to make it steeper, the other two would tend to make it harder. Detailed computation shows that the neutrino spectrum tends indeed to be slightly harder than the proton spectrum, due to the increase in the average number of neutrinos produced in \(pp\)  interactions with increasing energies (see e.g. [1]}, [2]}, [3]}).
Another potential implication of these results is that they affect the relation between the cutoff energy of the CR spectrum and the associated cutoff in the neutrino spectrum.
As an example of these effects, we show in Fig. REF  the total differential neutrino flux summed over all flavors (and multiplied by \(E_\nu ^2\) ), as well as the individual contributions from \(\nu _\mu ^{(1)}\)  and \(\nu _\mu ^{(2)}\) , resulting from an \(E^{-2}\)  CR proton spectrum with a sharp cutoff at \(E_c=100\)  PeV (left panel) and one with a smooth cutoff of the form d\(N_p/{\rm d}E\propto E^{-2}/\cosh ((E/E_c)^\delta )\) , with \(\delta =1\)  (right panel).We use the hyperbolic cosine suppression rather than an exponential one because with it the spectrum below \(E_c\)  is closer to the power-law \(E^{-\alpha }\)  and the determination of the hardening becomes clearer. One can see that the neutrino spectrum is harder than the original proton spectrum and it has a very smooth suppression starting at about 1 PeV. A good fit to the total neutrino spectrum can be obtained with the expression d\(N_\nu /{\rm d}E\propto E^{-\gamma }/\cosh ((E/E_\star )^{\delta ^{\prime }})\) , as shown in the plot. We obtain \(\gamma \simeq 1.91\) , \(E_\star \simeq E_c/107\)  and \(\delta ^{\prime }\simeq 0.45\)  for the sharp cutoff case (left panel), while \(\gamma \simeq 1.90\) , \(E_\star \simeq E_c/107\)  and \(\delta ^{\prime }\simeq 0.39\)  for the softer cutoff case with \(\delta =1\)  (right panel). Note that due to the wide spread of the energies of the neutrinos produced in \(pp\)  interactions, one indeed expects that the falloff in the neutrino spectrum should be significantly less pronounced than that of the original CR spectrum. It is worth noting that these results cannot be obtained if one computes the neutrino production in the so-called delta function approximation, i.e. setting the pion and neutrino energies in a \(pp\)  interaction to their average values (see e.g. [4]}).
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